Adaptive Control of HVDC Transmission Systems--Designing a stability-preserving energy-balancing outer loop

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Adaptive Control of HVDC Transmission Systems

Designing a stability-preserving energy-balancing outer loop

June 2021

Master's thesis

Yeray González

Arkaitz Rabanal

2021Yeray González, Arkaitz Rabanal NTNU Norwegian University of Science and Technology Faculty of Information Technology and Electrical Engineering Department of Electric Power Engineering

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Adaptive Control of HVDC Transmission Systems

Designing a stability-preserving energy-balancing outer loop

Yeray González Arkaitz Rabanal

Master in Renewable Energy in the Marine Environment Submission date: June 2021

Supervisor: Gilbert Bergna-Díaz

Co-supervisor: Raymundo E. Torres-Olguin

Norwegian University of Science and Technology Department of Electric Power Engineering

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To our supervisor, Professor Gilbert Bergna-D´ıaz, for his support, dedication and endless willingness to help from the early beginning of the Master’s Thesis. Without your guidance and perseverance this work would have never been possible.

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List of Figures

1 Share of renewable energy in power generation in the Sustainable Development Scen-

ario, 2000-2030. . . 3

2 European supergrid. . . 4

3 Breakeven distance between a HVAC and HVDC line. . . 5

4 (a) Π and (b) T models for a transmission grid. . . 5

5 Block diagram of a PID-PBC. . . 14

6 Block diagram of a PI-PBC. . . 15

7 VSC model for grid forming control strategy. . . 19

8 VSC model for grid feeding control strategy. . . 21

9 Schematic representation of the FD-Π model. . . 22

10 Block-diagram of the proposed PI-PBC. . . 32

11 Interconnection between VSC and PI-PBC. . . 34

12 Schematic representation of EBBA outer-loop. . . 53

13 HVDC Simulink Model diagram. . . 55

14 VSC AC currents and DC voltage in an open-loop control. . . 57

15 Injected active and reactive powers into the grid in an open-loop control. . . 58

16 VSC AC currents and DC voltage with PBC strategy . . . 59

17 Systems Lyapunov function and its time-derivative . . . 60

18 VSC AC currents and DC voltage with PBC strategy and erroneus parameters . . 60

19 Active and reactive powers with PBC strategy and erroneus parameters . . . 61

20 VSC AC currents and DC voltage with I&I philosophy forIT estimation. . . 62

21 I_T estimation. . . 63

22 VSC AC currents and DC voltage with I&I philosophy for G estimation. . . 64

23 Conductance estimation for a 10% of initial error. . . 65

24 VSC AC currents and DC voltage with I&I philosophy forIT and G estimation. . 66

25 Transmission current and conductance estimation for a 10% of initial error. . . 66

26 VSC AC currents and DC voltage with I&I philosophy for R estimation from one equation. . . 67

27 Inductor resistance estimation for a 10% of initial error. . . 68

28 VSC AC currents and DC voltage with I&I philosophy for R estimation from two equations. . . 69

29 Inductor resistance estimation for a 10% of initial error. . . 69

30 VSC AC currents and DC voltage with I&I philosophy forRandGestimation. . . 70

31 R andGestimations. . . 71

32 R andGestimations for different inductance and capacitance error. . . 71

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33 Zoom ofR andGestimations for different inductance and capacitance error. . . . 72

34 VSC AC currents and DC voltage with I&I philosophy forRandGestimation and for different inductance and capacitance error. . . 72

35 Zoom of VSC AC currents and DC voltage with I&I philosophy forRandGestim- ation and for different inductance and capacitance error. . . 73

36 VSC AC currents and DC voltage with I&I philosophy for R and G estimation and for different inductance and capacitance error (Grid feeder control with DC transmission grid). . . 74

37 Zoom of R andGestimations for different inductance and capacitance error (Grid feeder control with DC transmission grid). . . 74

38 Both terminals AC currents and DC voltages. . . 76

39 DC transmission line current (Reference from terminal 2 to 1). . . 76

40 Zoom of both terminals DC voltages and DC transmission line current. . . 77

41 Both terminalsR andGestimations. . . 77

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List of Tables

1 Electrical Nominal Values of the System . . . 56

2 Converter Passive Elements (per phase) . . . 56

3 PI-PBC proportional and integral parameters . . . 58

4 I_T estimator parameters . . . 62

5 Gestimator parameters . . . 64

6 Estimator parameters of DC current and conductance . . . 65

7 R estimator parameters. . . 68

8 AC resistance and DC conductance estimator parameters . . . 70

9 Terminals PI-PBC and EBBA controler parameters . . . 75

10 Terminal references though the simulation . . . 75

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ABSTRACT

The large-scale introduction of offshore renewable generation will indeed require a multi-terminal HVDC transmission system—based on voltage source (power) converters—in order to efficiently transfer the offshore power to shore over long distances. The conventional power converter control solutions associated to this technology (typically based on standard current control methods) have been designed to satisfactory operate near a single nominal operating point. This makes them inherently unsuitable for guaranteeing uninterrupted operation (stability) of the system in the event of an unexpected large signal-disturbance. Moreover, the offshore HVDC grid architecture complexity is expected to increase as more lines and renewable energy sources are connected to the grid. It then seems desirable to find control alternatives able to operate the system with large- signal stability guarantees, preserved regardless of any topological change or system disturbance size—and therefore avoiding any costly operational interruption of the system.

This manuscript presents theEnergy Balance Based Adaptive (EBBA)outer-loop rooted in theIm- mersion&Invariancemethodology, which replaces the traditional outer-loops while preserving the large-signal stability certificate on 2 Level VSCs. This novel controller places particular emphasis on the estimation of the system parameters, crucial for the converter load-flow computation. Ad- ditionally, for the development of this Master’s Thesisport-Hamiltonian representation,Lyapunov theory andPassivity Based Control strategy are used.

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1 INTRODUCTION

1.1 General Introduction

The current development and growing trends are expected to have huge impact on the global primary energy consumption over the next few decades. According to the International Energy Outlook 2019 [41], the U.S. Energy Information Administration (EIA) expects a growth of nearly 50% between 2018 and 2050; influenced by the strong economic development in Asia. Closely related to it, as consequence of the growth in end-use consumption, the electricity generation is expected to increase a 79% from the same time period. This can be interpreted as a direct consequence of the rising population and living standards in non-OECD countries and the electrification of the transportation sector, especially due to the plug-in vehicles and rail.

However, the way in which this increase will occur is not independent of the actual climate scenario.

For that reason, several global institutions have set forth ambitious energy plans; betting strongly on a massive inclusion of renewable energy sources (RES) in the energy system. This is the case of the objectives for 2030 and 2050 proposed by the European Union (EU), when the RE are expected to provide the 32% and 100% of the electricity, respectively [14, 33]. The impact of these kind of legislation may imply a worldwide renewable energy consumption increase of 3.1% per year. As a consequence of this transition trend, the EIA predicts renewable energy will be the most used energy source by mid 2040s decade. Figure 1 depicts the predicted share of renewable energy in the power generation according to the sustainable development scenario publish by the International Energy Agency (IEA) in 2020 [18].

2000 2005 2010 2015 2020 2025 2030

Year 15

20 25 30 35 40 45 50

Share of RES [%]

Figure 1: Share of renewable energy in power generation in the Sustainable Development Scenario, 2000-2030.

But, at the same time, this energy transition poses different challenges, which agents involved have to be able to solve. One of those challenges is the creation of efficient energy interconnections among large areas, including neighbouring countries and regions. This crucial step will contribute to reduce the dependence on the local predominant RE resource; i.e., a large enough grid including European and northern-African countries would count on the wind resource of the North Sea and the photovoltaic one of the countries of the south, among which the commissioned solar thermal projects in the north of Africa stand out. This kind of systems are known as supergrids (example shown in figure 1), which shares trans-national resources and, consequently, enormous RE potential.

Indeed, the implementation of supergrids, such as the one interconnecting Europe and North Africa shown in figure 2, and a further progress in the energy transition goes hand in hand with the development of technology that supports it [34]. This is the case of High Voltage Direct Current (HVDC) technology, which is probably the key technology for integrating large-scale RES. This kind of transmission system enables bulk energy transmission along great distances and the interconnection of different frequency HVAC grids. This is crucial as lots of RES are in remote locations, such as large water reservoirs or off-shore wind farms far away from the centres of consumption.

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Figure 2: European supergrid.

The use of HVDC technology took off in the mid 1950s, thanks to research on high power electronics and advanced electronic control in the two previous decades. Major advances in the field enabled the commissioning of the first commercial transmission link in 1954 in the middle of the Baltic Sea, feeding the Swedish island of Gotland. Furthermore, it is of interest to mention that this link is still operational, focused on transmission of wind power to mainland [2].

Although historically HVAC technology was predominant, the development of the mercury arc valves made the use of HVDC systems increase rapidly during the following years [35]. At the same time, in use voltage levels rapidly augmented from values of 110 kV to five or seven times the initial one. Such had been the interest in high voltage developing, that a transmission project of 1150 kV was built with testing purposes in the former Soviet Union.

The continuous development of AC transformers was the main reason of HVAC being historically the predominant transmission technology, reason why the terminal costs of this kind of installations are considerably lower at present. Nevertheless, its major drawback is the excessive reactive current drawn by the cable’s reactive element. This is directly related to the increase of cable’s losses as the line length is incremented. For that reason, reactive compensation technology is required, generally known asflexible AC transmission systems (FACTS). However, these elements augment the AC line cost per unit length.

On the other hand, recently introduced HVDC technology makes the energy transmission more cost-effective for longer distances, as the reactive losses are neglected in this state. Moreover, this scheme adds some other benefits such as the use of submarine and underground cables over long distances, relatively easy full control of the power flow, embedded stability, smaller footprint or reduced magnetic fields when compared to AC lines. Nonetheless, as the technology development is more recent, the constructing costs are considerably higher.

As a result, when discussing the usage of HVDC transmission systems, the most determining factor to consider is the so-calledbreakeven distance, which directly implies the economic feasibility of HVDC projects compared with HVAC [22]. For that reason, particular studies have to be done for every single case, as different technical and economic specifications and combinations could affect the result, obtaining the particular distance for each of the cases. An illustrative example is shown in figure 3.

Going in-depth on HVDC transmission lines, their modelling should be undertaken considering the balance between accuracy and complexity. Π and T models have been traditionally used considering their relatively good performance in spite of their simplicity.

In figure 4, both possible modellings are shown, being the disposition of the different components the main existing difference. In T models of transmission line, the shunt capacitance is positioned entirely in the middle of the line, while half of line resistance and reactance are assumed to be at

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AC te rm inal cos ts - including grid trans form e rs

2xSSC*

Transmission distance Investment

costs

* SCC = Series and shunt compensation of AC lines - required for each section of the line DC te rm inal cos ts

Total AC cos ts

AC line cos ts Break-even distance

Total DC cos ts DC line cos ts

Figure 3: Breakeven distance between a HVAC and HVDC line.

each side of the shunt capacitance. On the other hand, Π model presents half of shunt capacitance on each of the terminals, also called sending and receiving ends; while the whole resistance and reactance is in the middle of both of them. One of the main differences is where the flowing of the capacitive charging current is located. In the case of T model, this flows through half of the transmission line, while in Π this is shared by the transmission and the receiving end [11, 43].

It is interesting to highlight the predominance of the use of cascaded-Π models for those lines with great length, whose capacitance has to be properly distributed. However, such models fail to capture the frequency-dependent characteristics of the cable, which are essential to account for the damping effects of the system [6, 12].

In relation to this, a continuous development of transmission models has been crucial in order to capt an accurate representation of the HVDC transmission technology performance, specially when talking about cable-based connections over long distances. This is due to the presence of significant inductance and stray capacitance in the system, which cause multiple LC resonance frequencies; being damped oscillations, over-voltages and instability due to interaction with power electronic converters and their control systems the major problems to be faced [13].

On the other side, HVDC converters are responsible of adapting the power from AC to DC side and vice verse, and therefore, the control strategies are implemented directly over these terminal. They can be classified into two main groups depending on the switching technology: Line Commutated Converters (LCCs) andVoltage Source Converters (VSCs).

Line Conmutated Converters, which were used in early designs, are generally thyristor based technology; reason why they can be referred as “classic HVDC systems”. This is because self- conmutated power devices were unavailable in the early stages of the technology development.

Z/2 Z/2

Y Z

Y/2 Y/2

(a) (b)

Figure 4: (a) Π and (b) T models for a transmission grid.

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Due to the switching device’s nature, the current’s direction needs to remain invariant, so DC polarity needs to be changed if the power direction is wanted to be reversed. The major LCCs’

advantage are the lower station losses and costs, which enable higher converter ratings.

However, there is a major drawback for this technology, which is the high reactive power consumption at terminals. This can vary up to 50-60% of the rated reactive power. LCCs are generally applied when there is a highShort Circuit Ratio (SCR), i.e. when the grid is strong enough.

On the other hand,Voltage Source Converters appear as a relatively recent HVDC system, fun- damentally based on the development of self-commutated devices. These are mainly theinsulated gate bipolar transistor (IGBT), which combines the controllability of MOSFET and the reliability of BJT.

If this technology is compared to LCC, the size is drastically reduced, as no reactive power com- pensators or AC filters are required in this case. However, the losses increase when compared to the traditional technology. For a two level converter, losses can reach a 1.4%, while in LCCs it is of 0.8%; thus, the power ratings are lower. Even so, VSCs are the major trend when talking about HVDC systems, as they can provide more accurate control and resulting waves.

The commutation control of this technology is generally based onPulse Width Modulation (PWM) control; aiming to produce the most accurate sinusoidal AC voltage wave from DC and vice versa.

This is perfectly feasible with the primary converters, called 2-level VSC. Still, this typologies limit the resulting voltages and add harmonic content whenever the required RMS voltage differ substantially from the DC voltage. For that reason,Modular Multilevel Converters(MMCs), which is a natural evolution of 2-level converter, appear to be a fitting solution for obtaining better-quality AC signals. This is because of the addition of a higher number of switches, helping to move towards a large number of voltage levels. Consequently, the switching frequency of each module is reduced, and so does the number of filters and switching losses.

The traditionally used control for these VSC converters consists on two main loops: an inner-loop responsible for carrying system’s currents to their respective references, and an outer-loop which computes the needed current references based on the given active and reactive powers and voltages references.

Controlled currents are usually represented in a non-stationary dq0-framework, becoming these currents constant during steady-state operation. Thanks to this feature, traditionalProportional- Integral-Derivative (PID) controllers are able to bring the error between the measurements and references to zero.

In case of the inner loop, both direct- and quadrature-currents’ PID controllers must be tuned in order to cancel the dominant poles of the converter’s dynamic, allowing an appropriate control of the VSC behaviour. The dominant pole will be defined by the RL filter between the converter and the grid; thus, the time constantτ =_R^L should be canceled.

Conversely, the outer-loop generates a d-current reference from a previously requested active power or DC-side voltages. Taking into account that the DC voltage dynamic is dependant on the injected or absorbed AC active power, a single converter cannot control both at the same time.

The quadrature current reference is also computed by an outer-loop. In this case the q-current is controlled to get the asked reactive power or PCC’s voltage level, since both of them are strongly coupled in medium and high voltage AC systems.

Note that, to be able to neglect the inner-loop in the tuning of the outer-loop’s parameters, the former must be sufficiently faster than the latter.

Nevertheless, the global stability of these control strategy have not yet been proven analytically.

Thus, multiple numerical simulation must be carried out in order to be sure of the robustness and stability of the system. This fact takes even more relevance in multi-terminal HVDC systems, where the integration of new components must be carefully analysed aiming not to compromise the whole system.

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Consequently, the main motivation behind our work is to contribute towards the design ofrobust controllers with large-signal stability guaranteesable to suitably operate HVDC systems even when subjected to unwanted large disturbances. At the same time, particular consideration is given to control alternatives with Plug & Play philosophy, which would enable electrical grids to grow without requiring a significant update of the controllers, while preserving the overall stability certificate.

For that purpose, the passivity feature of the system will be exploited by means of aProportional- Integral Passivity Based Control (PI-PBC). This control philosophy enables an energetic interpretation of the control, which allows the mathematical interconnection of the physical and virtual systems, i.e. the converter and the control. Nevertheless, this strategy requires the knowledge of the desired equilibrium point of the system, which, in turn, requires the knowledge of the exact value of the system parameters.

Towards this end, the implementation of theImmersion & Invariance (I&I) methodology appears to be a promising solution, enabling to obtain the exact parameters that can guarantee the desired equilibrium point and which does not destroy the stability certification that has been previously obtained thanks to Lyapunov theory and thePBC strategy.

1.2 Objectives

This work aims to develop an analytical adaptive controller for non-linear systems, specifically focused on HVDC converters and systems, trying to improve the possible lack or wrong knowledge of system parameters. This will be particularly based onPassivity-Based Control and theImmersion

& Invariance methodology, which are explained in due course.

Accordingly, this present work has different objectives, which are identified as follows.

• First, we aim to obtain a suitable analytical nonlinear (incremental) model for a single 2- Level Voltage Source Converter (VSC), in both grid-feeding and grid-forming configurations, under the Port-Hamiltonian formalism, which eases the PBC design.

• Second, we aim to overcome the lack of knowledge over the physical system by means of the design of a parameters estimator, based on the immersion & invariance methodology.

• Finally, to validate the proposed adaptive controller through numerical simulation, testing both grid forming and grid feeding configurations, and increasing the complexity of the model.

In order to achieve all the objectives, and although this work is fed by the contributions from several references in the literature, the articleAdaptive PI Stabilization of Switched Power Converters[17]

of Dr. Hernandez-Gomez deserves special mention, since it provided the main idea for the project.

Likewise, the recent paperPID passivity-based Control of Power Converters: Large-signal Stability, Robustness and Performanceby Daniele Zonetti brought the basis for the understanding of PI-PBC strategy on VSC converters.

Despite the infinite research possibilities in this field, it is always important to take into consideration the limited time available for this Master Thesis development. For this reason, this work is focused on very specific objectives. Notwithstanding all of this, all possible complementary improvements are not discarded at all and they are included in the following section 1.3.

1.3 Limitation of Scope

As mentioned above, the current work has been developed in less than half a year. For that reason, different simplifications have been assumed so that the main objectives of the Master’s Thesis can be achieved. These assumptions are covered deeper in the following lines.

Firstly, it is well known that the modular multilevel converter (MMC) technology has established itself as the most suitable technology fot multi-terminal HVDC systems. However, for simplicity

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reason and time limitations, we focus our attention to its predecessor, the two-level voltage source converter (2L-VSC). This decision is justified given that i) the main focus of the work is on investigating the potential of the I&I adaptive control method for the HVDC scenario, and ii) it is likely that the results can be easily extended to the MMC.

Secondly, aiming to simplify as much as possible the model operation, the following assumptions are considered. On one hand, thecurrent inertia provided by the filter inductors is assumed to be large enough compared to the time constant of the switching action of the system so that the model is continuous. This assumes that both the modulation indices in theabc framework as well as the AC currents are considered purely sinusoidal. This decreases the relevance of current conditioning in relation to harmonics and quality.

On the other hand, the synchronisation of the converters to the network via PLLs is not taken into account. Again, this decision is justified given that i) the main objective is to highlight what the main contribution of the work is, rather than complementary features, and ii) the faster dynamic of the synchronisation process compared to the estimator ones makes this aspect possible to neglect to a certain extent.

Furthermore, we also assume that the transmission line parameters are known and available for load flow calculations. This strong assumption allows us to direct our focus to the estimations of the inner resistances and conductance of the power converters. Indeed, a more complete adaptive control design would also take the uncertainty of the line resistances into account, yet we leave this extension for future works.

Finally, decentralized and distributed control appear as the alternatives to facilitate the implementation of electrical systems in which the power flow is multi-directional. However, in a same way as in the previous point, a traditional centralized control scheme is assumed to simplify the system operation and estimator implementation. Again, alternative control modes are left for future improvements.

1.4 Thesis Structure

The remainder of this Thesis manuscript is structured as follows. Sections 2 and 3 follow parallel complementary paths: while the former gives the underlying theoretical framework necessary to understand the required concepts, the latter presents the application of such a framework to the specific case of the 2L-VSC. More precisely, section 2 presents a more formal approach, including the required theorems and lemmas; while in section 3 an informal interpretation is given, aiming to create a didactic summary that can serve as a basis for future master students not so familiar with nonlinear control to get acquainted with the necessary knowledge of this line of research.

Furthermore, section 4 is dedicated specifically to the I&I philosophy. More precisely, this section describes, in chronological order, our five different I&I implementation attempts, varying both the parameters to be estimated as well as the strategy undertaken. Ultimately, these attempts lead to our main result: a large-signal stability preserving outer-loop control based on the internal power- balance of the 2L-VSC. In section 5, different simulations results are presented, together with an analysis and discussion of the different estimators performance. Section 6 concludes the Thesis along with some hints for future works.

Finally, an appendix is attached in the last pages of the document, in which proofs, mathematical developments and theoretic background are included. This is since to some of them are well- known by people in the field or whose inclusion in the main document makes no sense for reasons of simplicity or excessive length.

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2 NON-LINEAR CONTROL THEORY PRELIMINARIES

This chapter will provide some basic yet formal theoretical concepts in order to ease the understanding for potential readers not so familiarized with nonlinear control. Starting with the definition of the equilibrium point and with the open loop control solution of a non-linear system, this section will introduce the concepts ofPort-Hamiltonian Representation,Nonlinear Incremental Model,Lyapunov Function,Passivity Based Control and theImmersion & Invariance Method.

It is worth mentioning that these concepts will be approached from a more theoretical point of view; leaving the application to the VSC to section 3.

2.1 Non-linear System Equilibrium

A physical system is said to be at equilibrium when the state-variables which represent the system are constant with respect to time. For instance, if the dynamics of a general system are expressed as in equation (1), wherex∈Rⁿ are the state-variables and ˙x=^dx_dt, there is an equilibrium point at ¯x∈Rⁿ that can be written as the set of assignable equilibrium (2).

˙

x=f(x) (1)

E :={x∈Rⁿ:f(x) =f(¯x) = 0} (2)

With some loss of generality, we will be focusing on non-linear systems whose dynamic equations can be expressed as in (3), whereu∈R^mis the control input vector of the system, the vector field f(x) :Rⁿ→Rⁿ and the input matrixg(x) :R^m→Rⁿ [50].

˙

x=f(x) +g(x)u (3)

The set of equilibrium of this kind of systems cansometimesbe expressed as function of state variables only, avoiding the control or input vector in the expression. Then, the assignable equilibrium point of (3), characterized by

0 =f(¯x) +g(¯x)¯u, can be alternatively written as (4).

E:={x∈Rⁿ :g^⊥(x)f(x) = 0}, (4)

whereg^⊥(x) is a full-rank left annihilator ofg(x). On the other hand, the input vector’s solution at the equilibrium can be computed as a function of the states as shown below.

u(¯x) =−g⁺(¯x)f(¯x), (5)

whereg⁺(x) is the Moore-Penrose left pseudoinverse of the matrixg(x) shown in equation (6).

g⁺(x) = g^>(x)g(x)⁻¹

g^>(x) (6)

2.2 Port-Hamiltonian Representation

Depending on the representation, the dynamics of the same physical system of interest can be written in a variety of forms. The election of the representation depends not only on the system

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itself, but it is also chosen in order to facilitate the development of the model to achieve the desired result.

For instance, a well-known mathematical model is the state-space representation, which is able to represent multiple-input multiple-output (MIMO) systems that are described by first order differential equations. Forlinear systems, this representation is written by the set of equations (7).

(x˙ =Ax+Bu,

y=Cx+Du. (7)

wherex∈Rⁿ are the state variables of the system, ˙xare the first order time-derivatives of state variables,u∈R^mis called input or control vector,y∈R^q is called output vector,A∈R^n×n is the state matrix,B ∈R^n×mis called input matrix,C∈R^q×n is the output matrix andD∈R^q×mis called feed-forward matrix [20].

Similarly, a set of dynamic equations describing a nonlinear systems can be also arranged in multiple forms. Looking at literature [17, 29], a commonly used representation is:

(x˙ =f(x) +g(x)u,

y=h(x) +j(x)u. (8)

However, even though the above mathematical model could be used for the representation of a VSC converter, since HVDC lines are composed by multiple components such as transmission lines and two or more converters, a model which allows an easy interconnection of different systems is preferred. Thus, any port-based model would be advisable.

Port-Hamiltonian representation fulfills the needed characteristics, since it combines the views of the Hamiltonian mechanics and the theory of port-based modelling [40], which allows the representation of complex systems by the interconnection of simpler blocks. This provides a unified framework for the modeling of systems belonging to different physical domains [42].

The Hamiltonian formulation of classical mechanics is formalized in a geometric way. The underlying geometric structure of port-Hamiltonian systems is determined by the interconnection structure of the system. This motivated to consider Dirac structures instead of Poisson structures as in traditional Hamiltonian formulation. This enabled to define Hamiltonian systems with algebraic constraints [38].

The main advantages that this mathematical representation brings to the project are listed below:

• A composition of Dirac structures are again a Dirac structure. Thus, a power-conserving interconnection of port-Hamiltonian systems is again a port-Hamiltonian system [42], what is of great relevance for the conservation of the stability certification.

• It enables an easy interpretation of physical properties such as conservation laws and energy considerations, including the shaping of energy-storage and energy-dissipation.

• It allows the integration of control, since it admits the extension of physical models with virtual system components [42]. This feature is applied later in section 3.4.4.

For the sake of brevity, we limit our attention to the modelling of switched power converters and transmission lines via the port-Hamiltonian formalism. Thus, for our application of interest, the representation takes the form given in equation (9) [17, 50].

˙

x= J0+

m

X

i=1

Jiui+R

!

∇H(x) + G0+

m

X

i=1

Giui

!

E (9)

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whereJi are the interconnection matrices withi∈ {0,1, ..., m},uiis thei^th element of the control vector, R=R^> ≥ 0 is the dissipation matrix, the vectorE ∈Rⁿ contains the external sources andGi∈R^n×n represents the interconnections with external sources.

The interconnection matricesJi capture the power preserving topological changes of the system introduced by the switches. Consequently, these matrices are skew-symmetric, i.e. they follow the structure given in (10). An interesting property of this kind of matrix for latter developments is that if J is a skew-matrix and c ∈ Rⁿ is a column vector, thenc^>Jc = 0 . A proof of this properties can be found in appendix C.

J =−J^> (10)

The Hamiltonian H(x) is a scalar function describing the energy-storage of the system, e.g. the energy in the inductances and capacitors in case of a power converter (see equation (11)). The effort of the energy-storing elements (currents and voltages) is then expressed by the Hamiltonian gradient∇H=^∂H_∂x ∈Rⁿ [40].

H(x) =1

2x^>Qx, Q=Q^>>0 (11)

In the specific case of a 2L-VSC, which has just fixed sources without any switching,Gi matrix can be omitted from (9). Then, the result fromG0E is simplified to E, obtaining the reduced expression (12) [17]. The development of the converters model up to the equation below is later presented in section 3.

˙

x= J0+

m

X

i=1

Jiu_i+R

!

∇H(x) +E (12)

It is worth mentioning that physical systems that are able to be represented under the port- Hamiltonian formalism, fall into the broader class ofpassive systems, making the former a natural starting point for the passivity-based control design [40], a characteristic that was widely used in [50]. Additional details on the PBC control of interest will be explained in section 2.4 and is applied to a 2L-VSC converter in section 3.4.4. However, before being able to apply this kind of controller, the nonlinear incremental model must be understood.

2.2.1 Nonlinear Incremental Model

From a general perspective, the concept ofpassivity characterizes dynamical systems that cannot store more energy than the energy supplied to them, i.e. that cannot generate power [32]. From this definition, it can be assumed that a VSC converter is passive. These energy conservation principles can later be restated to design stable control laws, especially in the case of large and complex control systems—via Lyapunov’s direct method (section 2.3).

The idea behind PBC, which is itself rooted in Lyapunov theory, is based on bringing the stored energy of the system to its minimum. Unfortunately, the implementation of this strategy directly to the Hamiltonian energy function would bring the state variables to zero, practically switching-off the converter.

One possible way of applying this control strategy with a non-zero equilibrium point is to instead base the control design on theincremental model, whose state variables are the deviations between original variables and the desired equilibrium point called incremental variables, i.e. ˜(·) := (·)−(·)¯ [19].

This new model is obtained after subtracting the solution of the system for the particular case of the desired point of equilibrium (13) from the port-Hamiltonian model. In such a way, the energy-based representation is developed up to an expression containing a Hamiltonian term with

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the form of (14). Refer to section 3.4.2 for a step by step derivation of the incremental model of the 2L-VSC.

˙¯

x= 0 = J0+

m

X

i=1

Jiu¯_i+R

!

∇H(¯x) +E (13)

H(˜x) =1

2˜x^>Q˜x=1

2(x−x)¯ ^>Q(x−x)¯ (14) The new Hamiltonian function is then a quadratic function which has its minimum at the point of desired equilibrium, enabling its used as Lyapunov function for non-zero equilibrium point systems.

2.3 Lyapunov Function Theory

When studying dynamical systems, different types of stability problems are generally found, where the stability of equilibrium points stands out. As referred in section 2.2.1, thenonlinear incremental model can be combined with Lyapunov’s direct method for nonlinear autonomous systems to bring the stored energy to zero, reaching one of the mentioned equilibrium points and, consequently, forcing the system to be stable. This is further explained in the following lines.

Consider an autonomous nonlinear dynamical system:

˙

x=f(x), x∈Rⁿ (15)

where the function (this is, the vector field)f : Rⁿ → Rⁿ satisfies a sufficient condition for the existence and uniqueness of a solution. The solution of (15), starting fromxat t= 0 is called system trajectory. ¯x∈Rⁿ can be defined as an equilibrium solution of (15) iff(¯x) = 0. This means the equilibrium point is a solution that does not change in time and, hence, they are degenerated solution curves that do not move [36]. By translating the origin to the equilibrium point ¯x by means of incremental model, zero can be made an equilibrium point. For the sake of simplicity, henceforth the origin is assume as an equilibrium point of (15) [37, 46].

Once the concept of equilibrium points is defined, Lyapunov’s second method for stability can be used to prove theasymptotic stability of a system by finding an scalar functionV(x), wherexis the state variables vector andV(x) is calledLyapunov Function (LF). This function must satisfy the following conditions:

1.

V(x)>0 ∀ x6= 0 (16) 2.

V(0) = 0 (17)

3.

V(x) =˙ ∂V(x)

∂t = ∂V(x)

∂x

∂t

| {z }

Chain rule

=∇V ·f(x)<0 ∀ x6= 0 (18)

4.

V(0) = 0˙ (19)

If ˙V(x) is negative, the function will decrease along the trajectory of (15) passing throughx. A V(x) is positive definite ifV(0) = 0 andV(x)>0 for x6= 0 as shown in the first condition. It is positive semi-definite if it satisfies the weaker conditionV(x)>0 for x6= 0. A function V(x) is negative definite or semi-definite if−V(x) is positive definite or semi-definite respectively [21].

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However, different kind of stabilities can be distinguished depending on the region of attraction (ROA), this is, the domain of the states xfrom which the system converges to the equilibrium point or the “safe” subset of the state space in which the system tends to stability asymptotically [8, 25].

Firstly, ifV(x) is a Lyapunov function, then the equilibrium is Lyapunov stable.

When the Lyapunov-candidate-function (LCF) V(x) is locally positive definite and the time derivative of the LCF ˙V(x) is locally negative definite, following (20); for the the subset B, the equilibrium is proven to belocally asymptotically stable.

V(x)˙ <0 ∀ x∈ B \ {0} (20) It is also important to remark that the system could also be proven to be asymptotically stable even when ˙V(x)≤0 through LaSalle’s invariance principle.

Finally, if the LCF is globally positive definite and the time derivative of the LCF is globally negative definite, following (21); for all the state space the equilibrium is proven to be globally asymptotically stable.

V(x)˙ <0 ∀ x∈R \ {0} (21) Where the LCF is positive definite, or radially unbounded, when

kxk → ∞=⇒ V(x)→ ∞

Furthermore, it is important to highlight in support of what is explained in the following sections that if a system is asymptotically stable in the Lyapunov sense and the Lyapunov function is the sum of two positive definite functions, then the system can be represented as the feedback interconnection of two passive systems [24].

Considering the LF stabilities, the nonlinear incremental model can be used to make a change of variable or offsetting, where the wanted operating point is settled as the equilibrium point, as will be developed in section 3. Likewise, there is no a general technique for constructing a LCF, nevertheless, a good starting point of a Lyapunov function candidate for physical systems can be based on the stored energy.

2.4 Passivity Based Control

During the last decades Passivity Based Controllers (PBCs) emerged as a non-linear control method which successfully exploits the physical structure of the systems, increasing in popularity as a building block for controller design [32]. As mentioned in previous sections, passive power systems could be defined as non-energy generator structures, whose difference between input and output power is just dissipations or energy storage. As stated by Dr. Lozano in [24], “Passivity is a useful analysis tool in the sense that the results can easily be interpreted in terms of energy in the system”.

As PBC is generally based on the passive energy balance of the system, this strategy is suitable for the use of second method of Lyapunov theory as a stability certification [42]. Note that, as said in section 2.3, a good starting point for finding a Lyapunov function candidates for physical systems is the stored energy of the system. A proof of this can be found in [39], where a control is designed for the classical problem of stabilization of an inverted pendulum on a cart based on the passivity property of the system, using both a port-Hamiltonian representation of the model and the Lyapunov theory.

In the literature it can be found a variety of control architectures based on this method. Among others, in [44] passivity based a fractional-order PID (FoPID-PBC) is implemented in a hardware-

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in-loop experiment for a grid-connected PV inverter via energy reshaping. Interconnection and damping assignment PBC (IDA-PBC) is studied in [26] and [47], which allows shaping the energetic properties of one or interconnected multi-system in a power-preserving manner. In this strategy the partial differential equations (which describes the system’s behaviour) that must be solved are parameterized by three matrices related with the interconnection between the subsystems. These matrices can be seen as a dynamic couplings that allow the propagation of dissipation. Moreover, PBC strategy can be also applyied to traditional PID controller. In this case, the input of the controller is not an error between a variable reference and its measurement, but the passive output of the physical system. Due to this fact, the combination of PID-PBC and port-Hamiltonian systems is widely studied first in [48], and later in [9] and [10]. A block diagram of a PID-PBC is depicted in figure 5 along with its model in equation (22).

Kp

Ki

+ +

u y

Kd

.

+

..

Figure 5: Block diagram of a PID-PBC.

(ξ˙=u

y=K_pu+K_iξ+K_dξ¨ (22) From the first stages of the methodology, different research areas have analysed the possibilities of PBC strategy in their processes. [45] studied control designs for the stability of distributed chemical processes in one spatial dimension, joining the passivity property with the second law of thermodynamics. A passivity-based perspective is also used in [3], overcoming some challenges of position, torque and impedance control of flexible joint robots. Moreover, the author emphasises the robustness of this control with respect to uncertainties of the robot.

Even though this control philosophy is not the predominant option in the electric industry, several articles can be found in the literature implementing this control successfully in multiple fields.

For instance, the application of a PI-PBC is studied in [31], where a controller is designed for a permanent magnet synchronous machine based on passivity properties of the motor. Furthermore, Lyapunov theory is applied to ensure that the system is globally asymptotically stable. Another example is found in [15], where a classical PID and a PID-PBC strategy are both theoretically and experimentally compared in the control of velocity of DC motors. However, this methodology is increasingly being used in areas that are currently leading the way in progress, such as power electronics.

In this latter field, incremental passivity based control is combined with generalized PI observer for the control of a DC-DC converter in [16]. With this structure the author is able to estimate the time-varying uncertainties in the output voltage and inductor current, proving the robustness by means of experimental studies. More specifically on voltage source converters, the performance of a MMC controlled by a PI-PBC is analysed in [7].

Due to the wide research about the PI-PBC in the literature, and as the traditional PID is already rooted in the industry (what facilitates its possible future implementation) PI-PBC strategy has been chosen as control for the converters stabilization.

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Kp

Ki

+ +

u ^. y

Figure 6: Block diagram of a PI-PBC.

Figure 6 depicts the block diagram representation of the used controller along with its mathematical model in (23). However, this system could be also interpreted from an energetic point of view as a I-PBC controller which is interconnected to the 2L-VSC by means of a dissipative component represented by the proportional gain. This fact is later analysed in the mathematical development performed in section 3.4.4.

(ξ˙=u

y=Kpu+Kiξ (23)

2.5 Immersion & Invariance

Even though the given physical system could be stabilized via a PI-PBC, and this stability could be proven through Lyapunov theory, the control still would have to deal with some uncertainties in different parameters. Note that the use of incremental model forces the control to run a load flow of the system in order to compute a desired equilibrium point.

Since the model of physical systems suffer from parametric uncertainties, some small errors are inevitably introduced into the load flow, resulting in an erroneous calculated equilibrium point. The deviation of calculated point with the real equilibrium depends on the influence of each parameter in the final solution of the system.

In [50] it is proven that for current-controlled 2L-VSCs, if the PID-PBC is fed with a load flow solution near a real equilibrium point, the system will stabilize close to the desired real equilibrium and thus limiting deviations, but without exactly regulating the currents to the desired references.

Immersion & Invariance(I&I) adaptive control could be a solution to recover the desired operating conditions prior to this deviation.

The I&I philosophy adds a new term to the classical certainty-equivalent control law, which is designed to achieve adaptive stabilisation of non-linear systems with parametric uncertainties that do not rely on the assumption of linear parameterization. As it is said in [5], the role of this new term is to shape the manifold into which the adaptive system is immersed.

A variety of articles about the implementation of this philosophy are found in the literature. In [27] I&I technique is studied for the adaptive control of linear multi-variable systems with reduced knowledge of the high-frequency gain, resulting in Lyapunov functions which contain cross-terms between the parameter errors and plant states.

[1] applies the I&I method in the stabilization of the traditional cart and pendulum system, aiming to design a functional control law and pointing the necessity of solving the partial differential equation of the immersion condition. Moreover, the design proposed in this article can be also studied from the perspective of PBC philosophy.

In another field, I&I is used by [4] to claim the possibility of design a globally convergent speed observer for general mechanical systems with k non-holonomic constraints, which main issue is recast as a problem of rendering attractive and invariant a manifold defined in the extended state-

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space of the plant and the observer. The contribution is validated implementing the observer in two practical examples.

The I&I adaptive control is proposed by [23] for nonlinearly parameterized nonlinear systems, departing from the standard adaptive control and from the property of monotonicity. With this approach, the authors exploits the latter to obtain a useful strategy in the whole state-space.

Apart from the use of I&I technique for the stabilization of adaptive controllers and state observers of nonlinear systems, [30] also proves the viability of this strategy for orbital stabilization, validating their results with a simple example of a 3-phase DC-AC converter. Additionally, port-Hamiltonian representation is mentioned as a possible structure to develop a systematic procedure to apply I&I.

The method will be explained by means of a brief example with a single uncertain parameter taken from [5]. In case of a deeper insight into the topic is wanted, the reader is referred to the mentioned reference.

Consider a non-linear system represented in the form,

˙

x=f(x) +g(x)u,

where the functionsf(x) andg(x) depend on an unknown parameterθ∈R. Then, a new variable is defined asθ^E ∈R (equation (24)), representing the estimation of the unknown parameter, so that lim

t→∞θ^E=θ.

θ^E,β(x) +γ (24)

Adopting this I&I approach, the (non-robust) cancellation is avoided comparing to classical adaptive controller and it provides a means of shaping the dynamic response of the estimation error (eθ). This is defined as,

eθ=θ^E−θ

Since the unknown parameter is constant, the dynamics of estimation errors follows the dynamic of the estimation parameter, and hence, estimation error can be developed as (25).

˙

eθ= ˙θ^E−θ˙= ˙θ^E = ˙β(x) + ˙γ (25) By means of chain rule, the time-derivative ofβ(x) can be expressed as in (26), getting a way to introduce the dynamics of the physical system into the adaptive estimator.

˙

eθ= ˙β(x) + ˙γ=∂β(x)

∂x x˙+ ˙γ=β⁰(x) ˙x+ ˙γ (26) Let’s consider an example of a first-order nonlinear system with the form of equation (27).

˙

x=θx²+u (27)

Then, it can be introduced to the dynamic expression of the estimation’s error.

˙

e_θ=β⁰(x) θx²+u

+ ˙γ=β⁰(x) (θ^E−e_θ)x²+u + ˙γ

The first important step of I&I method is to define an update law ( ˙γ) which linearizes or simplifies the dynamic of the estimator. In this example it should be,

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˙

γ,−β⁰(x) θ^Ex²+u Obtaining the next expression for the estimator’s error dynamic.

˙

eθ=−β⁰(x)x²eθ

Finally, function β(x) must be defined in order to get an stable system. A possibility for this example could be equation (28), forλ >0. However, other options could be also chosen. In section 4 different methods for the definition of this function will be discussed, but in short, it could be said that the idea is to obtain a final dynamic expression that can be solved as a exponentially decaying function, such that the error between the estimated and real parameter asymptotically converges to zero.

β(x) =λx³

3 (28)

β⁰(x) =λx²

(30)

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3 ENERGY MODELLING AND PI-PBC CONTROL FOR A 2L-VSC HVDC SYSTEM

Along this chapter, the theory explained in section 2 will be applied to the case of a 2L-VSC converter and a HVDC transmission line. Firstly, the models of both systems will be described and their port-Hamiltonian representations will be developed. Once the models are defined, open- loop stability will be checked.

In section 3.4 different mathematical tools will be used in order to obtain a global stability certificate. The explanations will be carried out in a chronological manner, obtained results and conclusions will be presented at each step. Note that one of the aims of this section is to be as didactic as possible for future readers. First, second method of Lyapunov will be developed for VSC port-Hamiltonian representation. Afterwards, in order to fulfill all the requirements needed by the second Lyapunov method, an incremental model is designed and implemented. After applying the Lyapunov method again, a proportional-integral PBC (PI-PBC) will be designed to finally fulfill all Lyapunov conditions.

3.1 Systems Modelling

Every analytical study of a physical system starts with a mathematical modelling, in which the state-variables are identified and their dynamic behaviours are described. In this first section of the chapter the modelling of both the converter and the HVDC line is performed, presenting the chosen voltage and current references for each system and modifying equations to facilitate their use in future sections.

3.1.1 Model description of the 2L-VCS

The analysed converter in this project is the well-known 2L-VSC converter, which is able to apply the desired voltage at AC-side terminals by means of an arrangement of six IGBT semiconductors.

Considering the converter model shown in figure 7, the set of dynamic equations (29) governing its AC currents’ behaviour can be obtained by means of Kirchhoff ’s Voltage Law (KVL). It is assumed that the converter works as inverter so the direction of the current is as indicated in the figure.

IT

Va

VDC1

Vc

Vb

La

Lc

Lb

Ra

Rc

Rb ia

ib

ic

N

G

C G

Figure 7: VSC model for grid forming control strategy.











−va− d

dt(ψa)−Ria(t) +VQN+uavdc1= 0

−vb− d

dt(ψ_b)−Ri_b(t) +V_QN +u_bv_dc1= 0

−v_c− d

dt(ψ_c)−Ri_c(t) +V_QN+u_cv_dc1= 0

(29)

where vk(t), k ∈ {a, b, c}, is the AC input voltage, ψk(t) are each inductor’s flux linkage, R is a resistance in the input of the converter,ik(t) is the current through the inductors, VQN is the voltage between the neutral and the negative side of the DC-capacitor,uk is the modulation index

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of each phase andvdc1(t) is the capacitor’s DC side voltage. Note that although the modulation indices are actually Boolean objects representing the on/off state of the semiconductors, in the average model of the converter they can be handled as sinusoidal waves. Considering the different terms that appear in (29), the following definitions and properties must be taken into account.

L= ψk

ik

(30)

C= q_c vdc1

(31)

i_C=Cdv_dc1

dt (32)

iG =Gvdc1 (33)

whereLis the inductance of the AC-side RL linkage,Cis the DC-side capacitance,qcis the charge accumulated due to capacitances, and G is the DC-side conductance.

The dynamic of the capacitor charge charge can be expressed as in (34), using the Kirchhoff ’s Current Law (KCL), where (31), (32) and (33) are applied. Using vector/matrix notation, the final equation (35) is obtained—refer to appendix A for a detailed derivation.

i_au_a+i_bu_b+i_cu_c=I_T−i_C−i_G (34)

˙

q_c=I_T −[u_abc]^T[i_abc]−i_G (35) Since the dynamics above are time-variant, for simplicity, a common approach is to use time- invariant equations using Park ordirect-quadrature-zero) transform, where theabc instantaneous voltages and currents are translated into a synchronously rotating framework, and where the direct- axis is linked to the a-phase voltage. This reduces the complexity of the computations and control strategy, allowing the system operator to control the active and reactive components of the current independently; or in other words,d andq axis respectively.

In any case, this transformation is widely known in the context of electrical and control engineering, for this reason why it is not directly included in the work. Yet, for the interested reader, this procedure can be found in appendix A. Finally, the set of equations after applying the transform is presented in (36). Same equations can be written from the point of view of rectifier operation and as a function of inductor’s fluxes and capacitor’s charges, which can be checked in appendix B.











Li˙d=−Rid+ωLiq+udvdc1−Vd

Li˙_q =−Riq−ωLi_d+u_qv_dc1−V_q Cv˙_dc1=I_T−3

2(u_di_d+u_qi_q)−Gv_dc1

(36)

Considering the previous sets of equations, the variablesid, iq and vdc1 are associated to system state variables ψd, ψq and qc respectively through expressions (30) and (31), while ud and uq

are the control variables. Since the system is defined by three equations with five variables, the system’s operator is able to decide the reference values of two of the states variables. In function of which set of state variables have been chosen as reference, the converter is said to be controlled as grid forming or grid feeding operation mode.

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On the one hand, the grid forming converters perform inDC voltage control mode (also referred asdirect output voltage control) when the VSC is required to control the reactive power and the DC voltage level. Thus, the terminologyforming is used because this kind of converters defines the existing voltage of the HVDC grid. From the modelling point of view, the converter’s DC side can be modelled along with an ideal DC current source, where the transmission line parameters are not considered. This was observed in figure 7 [49].

On the other hand, grid feeding converters operate inPQ control mode (ordirect current control) when the converter is required to control both active and reactive power. Consequently, the terminology feeding is used, as this converters define the currents fed to the AC side. When modelling this converters, the converter’s DC side can no longer be modelled as a DC current source. In this case, by contrast, the transmission line parameters are taken into account, after which a grid forming converter is generally found.

Vt

R_T1

RT2

RT3 L_T1

LT2

L_T3

VDC2

IT

VDC1

Va

Vc Vb La

Lc Lb Ra

Rc Rb ia

ib

ic

N

G C G

Figure 8: VSC model for grid feeding control strategy.

The most elementary HVDC system would be formed by a transmission line and two converters, each of them adopting one of control strategies. This kind of systems are presumably stable, but the aim of this work is to assure it mathematically, creating precursors for future enhancements.

One of the future research lines would include the control strategies that ensure a global stability in multi-terminal systems.

3.1.2 Model description of the HVDC transmission line

A mentioned in chapter 1, traditionallyπ-representation is the predominant model used for transmission lines. However, in this Thesis the model presented in [6] is used in order to capture the frequency-dependent characteristics of the cable. Moreover, the representation used in [50] is implemented, since in [12] is argued that 3 parallel branches are sufficient to reproduce the frequency dependency of the series impedance of the cable in the frequency range typically considered in studies of control interactions. The model has been schematized and the current references are shown in figure 9.

After applying KVL over the model, the set of equations (37) is obtained, which describe the behaviour of the individual current at each parallel branch.









 LT1

di_T1

dt =−RT1iT1+vdc2−vdc1

L_T₂di_T2

dt =−R_T2i_T₂+v_dc2−v_dc1 LT3

diT3

dt =−RT3iT3+vdc2−vdc1

(37)

whereRT i and LT i are the theoretical resistances and inductances of each branch, and,vdc1 and vdc2 are both converters’ DC voltage respectively. Take into account that the capacitances and conductances represented at both sides of the transmission lines will be added to the converters,

Adaptive Control of HVDC Transmission Systems--Designing a stability-preserving energy-balancing outer loop

Adaptive Control of HVDC Transmission Systems

Designing a stability-preserving energy-balancing outer loop

Master's thesis

Yeray González

Arkaitz Rabanal

Adaptive Control of HVDC Transmission Systems

Yeray González Arkaitz Rabanal

Contents

List of Figures

List of Tables

ABSTRACT

1 INTRODUCTION

1.1 General Introduction

1.2 Objectives

1.3 Limitation of Scope

1.4 Thesis Structure

2 NON-LINEAR CONTROL THEORY PRELIMINARIES

2.1 Non-linear System Equilibrium

2.2 Port-Hamiltonian Representation

2.3 Lyapunov Function Theory

2.4 Passivity Based Control

2.5 Immersion & Invariance

3 ENERGY MODELLING AND PI-PBC CONTROL FOR A 2L-VSC HVDC SYSTEM

3.1 Systems Modelling